Application of Curve Fitting and Surface Fitting Tools for ... · PDF fileIt is the force per unit area exerted on a surface by the weight of air above that surface in the atmosphere

International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064

Index Copernicus Value (2013): 6.14 | Impact Factor (2013): 4.438

Volume 4 Issue 4, April 2015

www.ijsr.net Licensed Under Creative Commons Attribution CC BY

Application of Curve Fitting and Surface Fitting

Tools for High Leverage Points and Outliers of

Wind Speed Prediction

Prema V., Uma Rao K.

Dept. of EEE, R.V. College of Engineering, Mysore Road, Bangalore 560059, India

Abstract: Climate change is generally accepted as the greatest environmental challenge our world is facing today. Together with the

need to ensure long-term security of energy supply, it imposes an obligation on all of us to consider ways of reducing our carbon

footprint and sourcing more of our energy from renewable sources. Wind energy is one such source and this paper proposes a few

methodologies to predict the speed of wind. Multivariate regression model, curve fitting, surface fitting model and time series model are

implemented for the test data collected from a wind farm. The results obtained are then compared with the actual values available for

validation.

Keywords: multivariate regression; time series; wind speed; prediction; curve fitting; surface fitting.

1. Introduction

Intermittency of wind is the biggest challenge in

implementing wind-energy as a reliable autonomous source

of electric power. Energy crisis, global warming, depletion of

fossil fuels are the major factors looming the world today.

Adequate utilization of renewable energy sources like wind,

solar, biomass etc. proves to be the only alternative to

overcome these problems. Because of the indeterminate

nature and complementary behavior of wind and solar

energy, many researchers are putting their effort towards

hybrid energy systems. A hybrid energy system uses a

combination of sources like wind and solar along with a

battery and Diesel Generator set.

Optimal allocation of the available resources is one of the

challenges faced in the design of hybrid energy systems. For

a well-planned system, it is essential to have accurate

predictive models of the wind and solar sources. Wind

energy is directly dependent on the wind speed available at

that location and this speed is extremely erratic. Hence, a

model is needed for accurate prediction of wind speed. This

paper proposes a few predictive models for wind energy

using regression and curve fitting.

A wide variety of methods were proposed in the literature for

wind forecast [1]. Persistence method or the „Naïve predictor

is the simplest and fundamental prediction model. Physical

approach makes use of Numerical Weather Predictor model

such as global forecasting system; prediktor etc. Statistical

approaches include time series forecasting methods and

Artificial Neural Network models. A few hybrid models are

also proposed which used a combination of the above

mentioned methods. This paper proposes four different

models for wind forecast based on regression, curve fitting

and surface fitting.

The correlation between the physical parameters was

obtained by using the statistical data analysis tools. Linear

regression, nonlinear regression, curve fitting and surface

fitting models were developed. A time series model was also

implemented for the purpose of tracking the peak overshoots

and undershoots. Here, time was defined as a variable along

with temperature, pressure and relative humidity and the

previous value of the wind speed was used as the input.

Errors for the various models were also calculated.

2. Parameters Used

There are various variables on which wind speed depends

upon and this paper presents the statistical relationship

among these variables which impact the wind speed in a wind

farm in Bagalkote region in Karnataka using MATLAB. The

parameters are typical of any moderate tropical country. The

variables used are temperature, relative humidity and

atmospheric pressure.

2.1. Atmospheric Pressure (P)

It is the force per unit area exerted on a surface by the weight

of air above that surface in the atmosphere of Earth. In most

circumstances atmospheric pressure is closely approximated

by the hydrostatic pressure caused by the weight of air above

the measurement point. On a given plane, low-pressure areas

have less atmospheric mass above their location, whereas

high-pressure areas have more atmospheric mass above their

location. Likewise, as elevation increases, there is less

overlying atmospheric mass, so that atmospheric pressure

decreases with increasing altitude. The unit of atmospheric

pressure used in this paper is consistent throughout and it is

mm of Hg.

2.2. Relative Humidity (RH)

It is the ratio of the partial pressure of water vapor in an air-

water mixture to the saturated vapor pressure of water at a

prescribed temperature. The relative humidity of air depends

on temperature and the pressure of the system of interest.

There is a considerable degree of difference between the

relative humidity measured indoors and that measured

Paper ID: SUB153600 2585





outdoors in regions such as wind farms and those at altitudes

of the shaft locations. Relative humidity, being a ratio has no

unit for and the numbers used in this paper are expressed as a

percentage.

2.3. Temperature (T)

A temperature is a numerical measure of hot and cold. Its

measurement is by detection of heat radiation or particle

velocity or kinetic energy, or by the bulk behaviour of

a thermometric material. The kinetic theory indicates

the absolute temperature as proportional to the average

kinetic energy of the random microscopic motions of their

constituent microscopic particles such as electrons, atoms,

and molecules. The basic unit of temperature in

the International System of Units (SI) is the Kelvin. For

everyday applications, it is often convenient to use the

Celsius scale, in which 0°C corresponds very closely to

the freezing point of water and 100°C is its boiling point at

sea level.

2.4. Wind Speed or Wind Velocity (Ws)

Wind is caused by air moving from an area of high pressure

to low pressure. Wind speed affects weather forecasting,

aircraft and maritime operations, construction projects,

growth and metabolism rate of many plant species, and

countless other implications. It is now commonly measured

with an anemometer but can also be classified using the older

Beaufort scale which is based on people's observation of

specifically defined wind effects. The unit of wind speed

used in this paper is consistently meters per second.

3. Data Acquisition

The data used for the proposed work was collected from a

wind farm in Bagalkot, Karnataka. The recorded data

consists of values of Air Temperature, Relative Humidity,

pressure, wind speed and the wind direction measured for

different altitudes, for every minute over a period of six

months. For analysis, one particular altitude values were

considered. The choice is made based on the fact that wind

velocity is higher in upper strata of atmosphere thereby

having more potential to generate wind energy[6]

. The data in

Microsoft Excel format is imported to MATLAB for the

analysis or modelling.

4. Modelling for Wind Speed Prediction

In regression models, the forecast is expressed as a function

of certain parameters which influence the outcome.

Regression gives an explanatory model that relates output to

various inputs which facilitates a better understanding of the

situation and different combinations of inputs can be tested

and their effects on the forecast can be studied. The

forecasting parameter is the wind speed and the independent

parameters used are atmospheric pressure, relative humidity

and average temperature.

Linear and non-linear regression models are developed.

Linear regression model is implemented by restricting the

number of dependent variables to one, which is wind speed.

Multivariate linear regression is often found to be more

efficient and useful since it considers the effects of all the

parameters that may have a significant effect on the

dependent variable present. In non-linear regression, the

inputs or the independent variables can have a non-linear

relation with the wind speed.

Curve fitting tool in MATLAB is used to find the best fit

between wind speed and one of the input variables. The best

fit is chosen to develop the model. Surface fitting is another

technique used to find the best fit between wind speed and

two of the input variables. A similar model was developed. A

time series model is also developed which takes the previous

value of wind speed and time as the inputs.

4.1. Linear Regression model

The input parameters considered for the linear regression are

combinations of the measured quantities (Pressure,

Temperature and Relative Humidity). A number of models

have been tested and the best model has been chosen which is

shown in equation 1. The inputs of temperature, pressure,

relative humidity, product of temperature and relative

humidity, product of temperature and pressure were observed

to yield the best results. The plot is shown in Figure 1 WS= -30.079*T+ 0.1257*RH- 0.0353P- 0.006*T*RH + .044*T*P (1)

4.2. Non-linear Regression model

A non-linear relation between the various combinations of

inputs was modelled using regression tool. It was found from

the analysis of theoretical data that there is proportionality

between wind speed and square root of pressure. This

relationship was also considered in the model. Various

combinations were tried out and the best of the lot was

chosen. The model is shown in equation 2. The results are

shown in fig.2.

WS= 0.3143*T+ 0.0408*RH - 0.2633*sqrt(P) (2)

4.3. Curve Fitting model

To implement this model, first the relationship between wind

speed and pressure is derived. This is obtained by plotting

the scatter diagram between pressure and wind speed. The

best fit is plotted as shown in fig.3. There is additional option

for including weights in the data points. The model thus

obtained is shown in equation 3.

WS = -9.565e-010 *(P)^3.757+53.08 (3)

4.4. Surface Fitting Model

The inputs yielding best results were those of pressure and

relative humidity and the surface fit is as shown in Figure 5.

The output graph obtained was slightly different from that

obtained from the previous functions and is given by Figure

6. The difference noted was the increased correlation

between the inputs and lower values of wind speed. The

obtained model is given in equation 4.

WS= -1525 +13.35*RH+4.131*P+ 0.00332*(RH)2 -

0.01909*RH*P -0.00278*(P) (4)






Figure 1: Plot for linear regression model

Figure 2: Plot for non-linear regression model

Figure 4: Plot for curve fitting model

Figure 3: Scatter diagram for curve fitting

Figure 5: Scatter diagram for surface fittng

Figure 6: Plot for surface fitting model






4.5. Time Series Model

Even in the best fit, the generated curve matched the original

curve in phase alone but not in magnitude. Observing all the

results, it can be concluded that an open loop system will

never give the required correlation. Hence, feedback systems

were incorporated in order to account for the mismatched

magnitudes. Thus, time series models were implemented to

improve the response.

In time series modelling, time was defined as a new variable

along with temperature, pressure and relative humidity. The

present value was predicted with the help of previous value

and the error. The input vector „X‟ was defined for an

intercept along with temperature and pressure.

The output curve showed better correlation with the input.

But, the output had very large offshoots causing large errors.

The output was enhanced by providing error correction by

shifting the output waveform by trial and error. Figure 7

shows plot. The following is the equation obtained after

modelling using time series is given in eq: 5

WS(time+1) = WS(time) + 0.2018 * T - 2.0873 (5)

Figure 7: Plot for time series model

5. Results and Discussions

As seen, the predicted values form a graph that is following

the trend of the plot formed by the actual values. The ups and

downs are followed, but the numerical data differs in the case

of multivariate regression models. In the case of time series

model both trend and the magnitude matched to a greater

extent.

Table 1: Comparison of Errors

Model SSE

Model regress 3763.851

Model nlinfit 3796.608

Model cftool 4178.766

Model sftool 3746.625

Time series model 3568.12

Table 1 shows the values of sum of squared errors for the

various models implemented.

6. Conclusion

Five different models were implemented for the prediction of

wind speed. It can be observed from the results and

calculated errors that the time series model has the least sum

of squared errors taken for about 942 points. Hence the

model with feedback elements yielded the best result. The

time series element solved the magnitude mismatch problem

and predicted wind speed with the highest accuracy as can be

seen in the output.

References

[1] Saurabh S. Soman, Hamidreza Zareipour, Om Malik, and

Paras Mandal, “A Review of Wind Power and Wind

Speed Forecasting Methods With Different Time

Horizons”, North American Power Symposium (NAPS),

2010 pp 1- 8

[2] A. M. Foley, P. G. Leahy, A. Marvuglia, and E. J.

McKeogh, "Current methods and advances in forecasting

of wind power generation" Renewable Energy, vol. 37,

pp. 1-8, Jan 2012.

[3] Chatterjee, S., and A. S. Hadi. "Influential Observations,

High Leverage Points, and Outliers in Linear

Regression”, Statistical Science. Vol. 1, 1986, pp. 379–

416.

[4] Seber, G. A. F., and C. J. Wild. “Nonlinear Regression ".

Hoboken, NJ: Wiley-Interscience, 2003.

[5] DuMouchel, W. H., and F. L. O'Brien. “Integrating a

Robust Option into a Multiple Regression Computing

Environment." Computer Science and Statistics:

Proceedings of the21st Symposium on the Interface.

Alexandria, VA: American Statistical Association, 1989.

[6] Holland, P. W., and R. E. Welsch. "Robust Regression

Using Iteratively Reweighted Least-

Squares."Communications in Statistics: Theory and

Methods, A6, 1977, pp. 813–827

Author Profile Prema V. received her Masters degree in Power

Electronics from Visvesvaraya Technological

University in 2005 and Bachelor‟s degree in Electrical

& Electronics Engineering from Calicut University in

2001. She is currently working as Assistant Professor

in R.V.College of Engineering, Bangalore, India. Her research

interests include Power Electronics, Renewable Energy and Digital

Signal Processing.

Dr. Uma Rao K. was born on 10th March 1963. She

obtained her B.E in Electrical Engineering and M.E in

Power systems from University Visvesvaraya college

of Engineering, Bangalore and her Ph.D from Indian

Institute of Science, Bangalore. She is currently Professor,

department of Electrical & Electronics Engineering, R.V. College

of Engineering, Bangalore. Her research interests include FACTS,

Custom Power, Power Quality , renewable energy and technical

education.


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Application of Curve Fitting and Surface Fitting Tools for ... · PDF fileIt is the force per unit area exerted on a surface by the weight of air above that surface in the atmosphere